Introduction to linear algebra

Incluye índice

1 Vectors and Matrices -- 1.1 Vectors and Linear Combinations -- 1.2 Lengths and Angles from Dot Products -- 1.3 Matrices and Their Column Spaces -- 1.4 Matrix Multiplication AB and CR -- 2 Solving Linear Equations Ax = b -- 2.1 Elimination and Back Substitution -- 2.2 Elimination Matrices and Inverse Matrices -- 2.3 Matrix Computations and A = LU -- 2.4 Permutations and Transposes -- 3 The Four Fundamental Subspaces -- 3.1 Vector Spaces and Subspaces -- 3.2 Computing the Nullspace by Elimination:A=CR -- 3.3 The Complete Solution to Ax = b -- 3.4 Independence, Basis, and Dimension -- 3.5 Dimensions of the Four Subspaces -- 4 Orthogonality -- 4.1 Orthogonality of Vectors and Subspaces -- 4.2 Projections onto Lines and Subspaces -- 4.3 Least Squares Approximations -- 4.4 Orthonormal Bases and Gram-Schmidt -- 4.5 The Pseudoinverse of a Matrix -- 5 Determinants -- 5.1 3 by 3 Determinants and Cofactors -- 5.2 Computing and Using Determinants -- 5.3 Areas and Volumes by Determinants -- 6 Eigenvalues and Eigenvectors -- 6.1 Introduction to Eigenvalues : Ax = λx -- 6.2 Diagonalizing a Matrix -- 6.3 Symmetric Positive Definite Matrices -- 6.4 Complex Numbers and Vectors and Matrices -- 6.5 Solving Linear Differential Equations -- 7 The Singular Value Decomposition (SVD) -- 7.1 Singular Values and Singular Vectors -- 7.2 Image Processing by Linear Algebra -- 7.3 Principal Component Analysis (PCA by the SVD) -- 8 Linear Transformations -- 8.1 The Idea of a Linear Transformation -- 8.2 The Matrix of a Linear Transformation -- 8.3 The Search for a Good Basis -- 9 Linear Algebra in Optimization -- 9.1 Minimizing a Multivariable Function -- 9.2 Backpropagation and Stochastic Gradient Descent -- 9.3 Constraints, Lagrange Multipliers, Minimum Norms -- 9.4 Linear Programming, Game Theory, and Duality -- 10 Learning from Data -- 10.1 Piecewise Linear Learning Functions -- 10.2 Creating and Experimenting -- 10.3 Mean, Variance, and Covariance -- Appendix 1 The Ranks of AB and A + B -- Appendix 2 Matrix Factorizations -- Appendix 3 Counting Parameters in the Basic Factorizations -- Appendix 4 Codes and Algorithms for Numerical Linear Algebra -- Appendix 5 The Jordan Form of a Square Matrix -- Appendix 6 Tensors -- Appendix 7 The Condition Number of a Matrix Problem -- Appendix 8 Markov Matrices and Perron-Frobenius -- Appendix 9 Elimination and Factorization -- Appendix 10 Computer Graphics -- Index of Equations -- Index of Notations -- Index